Kenneth M. Levasseur

Factoring in ℤ[√ d ]

You should have an elementary understanding of divisors and factoring with integers. It may also be helpful to be familiar with the ring ℤ[√d].

Rotational Groups of Regular Polyhedra

To complete this lab, you should know how a group can be generated from a set of elements and a binary operation. You should also be familiar with Euler angles (see the Rotations Lab on the CD for a review) and group actions.

Is This a Group

To complete this lab, you should have already seen the definition of a group and become familiar with the basic group properties: being closed, having an identity, inverses, and associativity (and commutativity).

Quotient Rings of Polynomials

To complete this lab, you should be familiar with the ring of polynomials over a field, the division property for polynomials over a field, and the definitions of homomorphism, kernel, and ideal. Finally, you should be familiar with the First Isomorphism Theorem for ring homomorphisms (Ring Lab 5).

Quadratic Field Extensions

To complete this lab, you should be familiar with the construction of quotient rings of the ring of polynomials over a field F. You should also be familiar with irreducible polynomials over a field. This lab does not presume any other prior knowledge of field extensions. Doing Ring Lab 10 first would be helpful, but it is not necessary.

Let’s Get These Orders Straight

To complete this lab you should be familiar with the basic definition of a group. You should also be familiar with the definition of the order of an element in a group. (Recall that the order of an element g of a finite group G is the least positive integer k such that g k is equal to the identity of G.)

Determining the Symmetry Group of a Given Figure

Though not absolutely necessary, it would be useful if you completed Group Lab 1 before attempting this lab.

Introduction to Abstract Algebra

This guide is written with the assumption that the reader has at least minimal familiarity with groups, rings, and homomorphisms; consult an abstract algebra text for details of any unfamiliar algebraic concept. A bibliography in the Appendix contains some suggested references. The purpose of this guide is to provide details for (and illustrations of) many of the structures and functions used in the packages in AbstractAlgebra. Many of these structures and functions are also used in the laboratory notebooks in Exploring Abstract Algebra with Mathematica. For updates to this guide, updates to the packages, as well as other related resources, the web page http://www.central.edu/eaam.html (which is mirrored at http://www.uml.edu/Dept/Math/eaam/eaam.html) can be consulted.

Introduction to Rings and Ringoids

There are no prerequisites for this lab, although a brief introduction to the terminology related to rings might be beneficial.

What Does ℤ[i] / Look Like?

Prior to working on this lab, you should be familiar with the term ideal through discussions in class or from Ring Lab 3. You should also be familiar with an integral domain, a field, and the characteristic of a ring.